On the other hand, values of the filling factor \(f\)  close to unity pose significant challenges for particle acceleration models that involve highly localized geometries, such as super-Dreicer acceleration in thin current sheets [1]}, [2]}. Turning to the specific acceleration rate in such an acceleration scenario, [3]} find (their equation [3.17]) a rate of proton acceleration \(d{\cal N}_p/dt \simeq 2 \times 10^{37} \, \sqrt{\zeta }\)  s\(^{-1}\) , where \(\zeta \)  [3]} is the Lundquist number, the ratio of the diffusive to advective terms in the magnetic diffusion equation. Following [3]}, we take \(\zeta = 10^{-8}\) , giving a proton acceleration rate \(d{\cal N}_p/dt \sim 2 \times 10^{33}\)  s\(^{-1}\) , and we take the number of protons available for acceleration as \({\cal N}_p \sim \varepsilon n \, L^3\) , where \(\varepsilon \simeq 0.4 \, \sqrt{\zeta } \simeq 4 \times 10^{-5}\)  is the angle at the magnetic X-type neutral point at the origin of the acceleration region. With an ambient density \(n \simeq 10^{11}\)  cm\(^{-3}\)  and a longitudinal acceleration region extent \(L \sim 10^9\)  cm, \({\cal N}_p \sim 4 \times 10^{33}\)  and so \(\eta \simeq 2\)  s\(^{-1}\) . Although this value is much greater than the values of \(\eta \)  deduced here (it corresponds to the acceleration of all ambient particles in less than a second), it must be again stressed that the [3]} model refers to highly efficient acceleration of a relatively small number of protons in a very localized geometry. We encourage calculations of specific acceleration rates for electrons in such a model, and indeed in all theoretical particle acceleration scenarios.
